I will describe how non-invertible global symmetries act on operators in a quantum field theory. The various possible actions are called generalized charges. This provides a stepping stone for understanding physical applications of non-invertible symmetries, as will be exemplified in the case of Ising symmetry. One of the surprising findings of this endeavor is that there exist new and unexplored generalized charges already for ordinary invertible global symmetries! These generalized charges are described by higher-representations of the symmetry group, generalizing the ordinary charges described by ordinary representations of the symmetry group.

Relative theories are non-topological theories living at the boundaries of TQFTs in one higher dimension. An interesting and well-studied class of relative theories are 6d N=(2,0) theories. I will introduce the notion of relative defects in relative theories, which are non-topological defects of the relative theory living at the boundary of a topological defect of the above-mentioned TQFT in one higher dimension. I will argue that codimension two defects of 6d N=(2,0) theories are relative defects. Relative defects carry ''trapped'' higher-form symmetries localized on their world-volume which are independent from the higher-form symmetries of the bulk theory. When the bulk theory is compactified with the insertion of relative defects, the trapped higher-form symmetries provide extra contributions to the higher-form symmetries of the lower-dimensional theory resulting from the compactification. For example, when 6d N=(2,0) theories are compactified on a Riemann surfaces with punctures (which are relative codimension-two defects) then the 1-form symmetry of the resulting 4d N=2 Class S theory obtains contributions from the 1-form symmetries trapped at the punctures, along with the well-known contribution coming from the 2-form symmetry of the 6d N=(2,0) theory; part of London TQFT Journal Club; it will be possible to follow this talk on Zoom (please register at https://london-tqft.vercel.app)